Scale-Invariant Model of Boltzmann Statistical Mechanics and Generalized Thermodynamics

Abstract

Some implications of a scale-invariant model of Boltzmann statistical mechanics to the laws of generalized thermodynamics are investigated. Through definition of stochastic Planck and Boltzmann universal constants, dimension of Kelvin absolute temperature T (degrees kelvin) is identified as a length (meters) associated with Wien wavelength Tβ = λwβ of particle thermal oscillations. Hence, thermodynamic temperature and atomic mass of the field 𝔽β at scale β provide internal measures of (extension, duration) of background space 𝕊β+1 = 𝔽β needed to define external space and time coordinates and atomic-mass-unit of 𝔽β+1. Introduction of invariant internal thermodynamic spacetime and Boltzmann factor are in harmony with modern concepts of quantum gravity as deterministic dissipative dynamic system [73]. The connections between de Pretto number 8338 and Joule-Mayer mechanical equivalent of heat Jc = 4.169 kJ / kcal and universal gas constant Ro = 8338 J / kcal.m are identified leading to the modified mechanical equivalent of heat J = 2Jc = 8338 J / kcal . It is shown that with work defined as Helmholtz free energy W = F = U – TS, Helmholtz decomposition of total thermal energy into free heat U and latent heat pV results in modified form of the first law of thermodynamics Q = H = U- W = U + pV. Finally, by application of Boltzmann combinatorics method, entropy of ideal gas is expressed in terms of the number of Heisenberg-Kramers virtual oscillators as S = 4Nk in exact agreement with predicted entropy of black hole by Major and Setter [159].